We will pick things up just where we left off last week in Manuel's talk. We will discuss combinatorial and geometric models for manifold diagrams (i.e. higher dimensional generalizations of string diagrams) based on recent joint work with Chris Douglas. We focus on three aspects of the theory: (1) the geometric duality of manifold diagrams and pasting diagrams, whose cells provide a novel "universal" class of shapes for higher category theory; (2) how to extend the tantalizing connection between classical singularities and laws of dualizable objects into higher dimensions, overcoming obstructions faced by classical differential singularity theory; and (3) the conjectural "combinatorialization" of smooth structures, which would allow us to faithfully represent smooth structures of manifolds in manifold diagrams, and thus by purely combinatorial means.